where \(\gamma ^{\prime } = \frac{d\gamma }{dz}\) , while the remaining equations in (REF )-() are left unaltered. Two observations are in order. First, for the specific case of sum-rate utility, we have that \(\beta _i(z) = z\) . This implies that \(\gamma _i(z) = \log (z)\)  and it becomes apparent that () is recovered from (REF ).
Second, one can leverage [1]} to show that the optimal power allocation is a fixed point of our modified iteration for \(a_i^{(k)} = 1, b_i^{(k)} =0\)  under the condition that \(\gamma _i(z)\)  is strictly concave for \(z>0\) .
The empirical effect of the modification in (REF ) is studied in Section REF .

